1896.] certain Properties of the Tetrahedron. 99 



and therefore 



( Win) 2 = m . Km = (Um+ Vm) ( Um - Vm) = ( Um) 2 - ( Vm) 2 . 



Thus we obtain, for the characteristic equation of the matrix 

 Vm, the equation 



( Vm) 2 + ( Wm) 2 - ( Um) 2 = 0, 

 from which it follows that 



JJVm = 0, WVm = (( Wm) 2 - ( Unif}? (5). 



We also have 



(Km) 2 = ( Urn) 2 - 2 UmVm + ( Vm) 2 



= 2 ( Urn) 2 - 2 UmVm - ( Wm) 2 

 = 2 UmKm — ( Wm) 2 , 

 or (Km) 2 - 2 UmKm + ( Wm) 2 = 0, 



from which it follows that 



UKm= Urn, WKm= Wm (6). 



Now suppose that we have a second matrix m', whose character- 

 istic equation is 



m 2 - 2\W + // = (7). 



Then, in addition to equations (1) and (7), we have an equation 

 involving both m and m', of the form 



mm' + mm — 2\'m — 2\m' + 2v = (8). 



The equations (1), (7), (8) are said to constitute the catena of 

 relations satisfied by m and m'. 



We will introduce one more symbol L, defined by the equations 

 Lmm! = Lm'm = v (9). 



We have 



mKm' 4- m'Km = m(2 Um — m) + m (2 Um — m) 

 = 2m Um' + 2m Um — (mm' + m'm) 



= 2Lmm (10), 



by means of equation (8). 



In the sequel we shall sometimes be concerned with the matrix 

 unity. When it is necessary to bring this matrix in evidence wf 

 shall denote it by the symbol to, otherwise we shall treat it in tb 

 customary manner. The said matrix has both its latent root, 

 equal to unity, and thus we shall have 



U(o = l, Wa> = l, Kco = (o, V<o = 0, 

 Lmoo = Loom = Um. 



7—2 



